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Nonstandard finite difference models of differential equations. (English) Zbl 0810.65083

Singapore: World Scientific. xi, 249 p. (1994).
The aim of this book is to find finite difference schemes for the numerical integration of differential equations for which the true solution is an exact solution of the discrete equations or, when this is not achievable, the qualitative behaviour of the exact and discrete solutions are similar. In particular the two solutions should satisfy the same stability properties and the finite difference scheme should not have instabilities that are not present in the continuous solution.
Chapter 1 presents standard finite difference schemes for some simple ordinary and partial differential equations.
Chapter 2 discusses numerical instability that occurs when the step exceeds a certain size, or in some cases when the step exceeds zero. There is a discussion of schemes for the one-way wave equation which shows that the exact solution may satisfy the finite difference scheme. Numerical stability is not analysed.
Chapter 3 introduces some nonstandard finite difference schemes, pointing out that in the case of a linear differential equation of order \(n\) the \(n\) linearly independent solutions may be used to find an exact linear difference equation of order \(n\) which any solution of the differential equation satisfies exactly. This difference equation will not, of course, coincide with any of the standard finite difference schemes. After a derivation of several such exact schemes five rules for the construction of discrete models are presented and their application to a nonlinear oscillator and a nonlinear diffusion problem demonstrated.
Chapter 4 considers autonomous first-order equations with a view to matching the stability of the difference scheme to that of the differential equation in the neighbourhood of its fixed points. As with conventional finite difference methods this requires a knowledge of \(f_ y\) at the fixed points.
Chapter 5 looks at second-order nonlinear oscillators, and the possibility, for nonlinear perturbations of modelling the asymptotic expansion in the small parameter.
Chapter 6 looks at coupled pairs of first-order equations.
Chapter 7 treats partial differential equations with an emphasis on reproducing separable solutions. It is rather unspecific about the relation between space and time steps and the unwary reader might be led to choices of the two which resulted in instability or inconsistency with the original equation.
Chapter 8 discusses the Schrödinger equations starting with the simple equation \(u_ t = i u_{xx}\). The eigenvalues of the spatial derivative are purely imaginary and hence standard schemes are unconditionally unstable. Unfortunately the method proposed is consistent with \(u_ t = (1 + i)u_{xx}\).
Chapter 9 sums up the previous chapters and looks again at the rules presented in chapter 3. It concludes with two further examples of which one, the differential equation satisfied by the Weierstrass elliptic function \(\wp(z)\), illustrates both the interest and the limitations of the method: an exact discrete scheme exists, but it requires a knowledge of \(\wp(h)\), where \(h\) is the step.
The book does not discuss a posteriori error bounds or the application to general systems of differential equations. There are no numerical examples.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L12 Finite difference and finite volume methods for ordinary differential equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
34A34 Nonlinear ordinary differential equations and systems
35K55 Nonlinear parabolic equations
35L60 First-order nonlinear hyperbolic equations
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